Abstract
Modeling of wall-bounded turbulent flows is still an open problem in classical physics, with relatively slow progress in the last few decades beyond the log law, which only describes the intermediate region in wall-bounded turbulence, i.e., 30–50 y+ to 0.1–0.2 R+ in a pipe of radius R. Here, we propose a fundamentally new approach based on fractional calculus to model the entire mean velocity profile from the wall to the centerline of the pipe. Specifically, we represent the Reynolds stresses with a non-local fractional derivative of variable-order that decays with the distance from the wall. Surprisingly, we find that this variable fractional order has a universal form for all Reynolds numbers and for three different flow types, i.e., channel flow, Couette flow, and pipe flow. We first use existing databases from direct numerical simulations (DNSs) to lean the variable-order function and subsequently we test it against other DNS data and experimental measurements, including the Princeton superpipe experiments. Taken together, our findings reveal the continuous change in rate of turbulent diffusion from the wall as well as the strong nonlocality of turbulent interactions that intensify away from the wall. Moreover, we propose alternative formulations, including a divergence variable fractional (two-sided) model for turbulent flows. The total shear stress is represented by a two-sided symmetric variable fractional derivative. The numerical results show that this formulation can lead to smooth fractional-order profiles in the whole domain. This new model improves the one-sided model, which is considered in the half domain (wall to centerline) only. We use a finite difference method for solving the inverse problem, but we also introduce the fractional physics-informed neural network (fPINN) for solving the inverse and forward problems much more efficiently. In addition to the aforementioned fully-developed flows, we model turbulent boundary layers and discuss how the streamwise variation affects the universal curve.
Highlights
Reynolds [1] was the first to statistically describe turbulence by decomposing the instantaneous velocity vector into an average field and its fluctuation
As the experimental data were only available for y+ > 10,000, we synthesized an entire profile from the pipe wall to centerline using multifidelity Gaussian process regression (M-GPR) [29] as follows: we considered as high fidelity data the superpipe data in the outer region together with the highest direct numerical simulations (DNSs) data for channel flow at Reτ = 5200
We proposed multiple fractional models for wall-bounded turbulent flows in benchmark cases where the mean flow is either one-dimensional or two-dimensional
Summary
Reynolds [1] was the first to statistically describe turbulence by decomposing the instantaneous velocity vector into an average field and its fluctuation. Upon substitution into the Navier–Stokes equations and averaging, assuming quasi-stationarity, a new modified equation emerged for the average velocity that includes an additional term, namely, the averaged dissipation tensor leading to the turbulence-closure problem [2]. At about the same time, Richardson [4], in an attempt to unify turbulent diffusion with molecular diffusion, combined geophysical measurements with Brownian motion to produce the famous scaling law on turbulent pair diffusivity. While ingenious, both approaches assume implicitly locality in turbulent interactions, which limits the universality of the derived correlations—an open standing question for over a century. As stated by Kraichnan [5], Prandtl’s approach is valid only when the spatial scale of inhomogeneity of the mean field is large compared to the mixing length
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